Chapter 9: Signaling Over Bandlimited Channels EE456 – Digital Communications Professor Ha Nguyen September 2015 EE456 – Digital Communications 1
Chapter 9: Signaling Over Bandlimited Channels Introduction to Signaling Over Bandlimited Channels We have considered signal design and detection of signals transmitted over channels of “infinite” bandwidth, or at least sufficient bandwidth to pass most of the signal power (say 99% or 99 . 9% ). Strictly-bandlimited channels are common: twisted-pair wires, coax cables, etc. Band-limitation depends not only on the channel media but also on the source rate ( R s symbols/sec). If one keeps increasing the source rate, any channel becomes bandlimited. Band-limitation can also be imposed on a communication system by regulatory requirements. In all commercial communications standards, specific spectrum masks are placed on the transmitted signals. The general effect of band limitation on a transmitted signal of finite time duration is to disperse (or spread ) it out ⇒ Signal transmitted in a particular time slot interferes with signals in other time slots ⇒ causing inter-symbol interference (ISI). In this chapter, we shall consider signal design and demodulation of signals which are not only corrupted by AWGN, but also by ISI. Major Approaches to Deal with ISI: Force the ISI effect to zero ⇒ Nyquist’s first criterion. 1 Allow some ISI but in a controlled manner ⇒ Partial response signaling. 2 Live with the presence of ISI and design the best demodulation for the situation ⇒ 3 Maximum likelihood sequence estimation (Viterbi algorithm). EE456 – Digital Communications 2
Chapter 9: Signaling Over Bandlimited Channels Baseband Antipodal Communication System Model ✁ � ✂ � ✒ ✝ � ✠ ✎ ✏ ✡ ✂ ✖ � ✠ ✌✍ ✡ ✏ ✡ ✂ ✟ ✌ ☛ ✠ ✂ ✡ ☞ ✄ ✌� ✍ ✆ ✄ ☎✝ ✞ ✟ ✕ ✡ ✍ ✂ ✟ ✝ ✗ ✓ � ✠ ✠ ✟ ✍ r ( Rate ) H T ( f H C ( f ✑ ✄ ✌ ☎ ✍ � ✂ ✄ ✝ ) b ) ✖ ✡ ✠ � ✝ ✘ ✁ � ✂ � w t ( ) (WGN) t = kT b ✙ ✟ ✞ ✟ ✡ ✔ ✟ ✁ ✟ ✞ ✡ ✎ ✡ ✄ ✠ ✁ � ✂ � ✆ ✡ ✠ ✚ ✕ ✡ ✍ ✂ ✟ ✝ H R ( f ✁ ✟ ✔ ✡ ✞ ✟ ) Equivalent system model: w t ( ) 1 1 0 ✦ t = kT b T 2 ✢ ✣ ✤ ✢ ✣ ✤ b ✜ ✛ H C ( f H R ( f H T ( f ) ) ) T 0 r y t t b ( ) ( ) ✢ ✥ ✣ ✤ EE456 – Digital Communications 3
Chapter 9: Signaling Over Bandlimited Channels ISI Example: Modulation is NRZ-L and Channel is a Simple LPF R Lowpass Filter C t 1 h t = − ✧ ★ ( ) exp ✩ ✪ C RC RC ✫ ✬ ✭ ✮ s B ( t s A ( t ) 1 ) V V − − T RC (1 e / ) b RC t t T T 0 0 b b − s t − T − s t − T ✯ ✰ ✱ ✯ ✲ ✱ ( 2 ) ( 4 ) A b A b V T T T 3 4 5 b b b t T T 0 2 b b s A ( t s t − T s t − T ) ( ) ( 3 ) A b A b ✳ ✴ ✵ − − − s t − T s t T ( 4 ) ( 2 ) B b B b T 4 b t 0 T T T T 2 3 5 b b b b − s B ( t s t T s t − T ) ( ) ( 3 ) B b B b ✶ ✷✸ r b b b b w t = s t + s t − T + s t − T + s t − T + t During this interval, ( ) ( ) ( ) ( 2 ) ( 3 ) ( ) B B b B b B b 0 1 2 3 EE456 – Digital Communications 4
Chapter 9: Signaling Over Bandlimited Channels Baseband Message Signals with Different Pulse Shaping Filters Information bits or amplitude levels 1 0 −1 2 4 6 8 10 12 14 16 18 20 n Output of the transmit pulse shaping filter − Rectangular 2 0 −2 2 4 6 8 10 12 14 16 18 20 t / T b Output of the transmit pulse shaping filter − Half−sine 2 0 −2 2 4 6 8 10 12 14 16 18 20 t / T b Output of the transmit pulse shaping filter − SRRC ( β =0.5) 2 0 −2 2 4 6 8 10 12 14 16 18 20 t / T b Spectrum of the transmitted signal is much more compact in frequency if the impulse response of the pulse shaping filter is made longer in time. The signal corresponding to one bit occupies a duration that is longer than one bit interval, hence causing interference to signals of adjacent bits (inter-symbol interference, or ISI). It is theoretically possible to design the pulse shaping filter so that the effect of ISI at the sampling moments is zero! EE456 – Digital Communications 5
Chapter 9: Signaling Over Bandlimited Channels Nyquist Criterion for Zero ISI w t ( ) 1 1 0 ✿ t = kT b T 2 ✻ ✼ ✽ ✻ ✼ ✽ b ✺ ✹ H C ( f H T ( f H R ( f ) ) ) T 0 r y t t b ( ) ( ) ✻ ✾ ✼ ✽ ∞ � y ( t ) = b k s R ( t − kT b ) + w o ( t ) , k = −∞ where s R ( t ) = h T ( t ) ∗ h C ( t ) ∗ h R ( t ) is the overall response of the system due to a unit impulse at the input. � V if the k th bit is “1” b k = if the k th bit is “0” . − V Normalize s R (0) = 1 and look at sampling time t = mT b : ∞ � y ( mT b ) = b m + b k s R ( mT b − kT b ) + w o ( mT b ) . k = −∞ k � = m � �� � ISI term What are the conditions on the overall impulse response s R ( t ) , or the overall transfer function S R ( f ) = H T ( f ) H C ( f ) H R ( f ) which would make ISI term zero? EE456 – Digital Communications 6
Chapter 9: Signaling Over Bandlimited Channels Time-Domain Nyquist’s Criterion for Zero ISI The samples of s R ( t ) should equal to 1 at t = 0 and zero at all other sampling times kT b ( k � = 0) . Im pulse applied t = at 0 ❀ ❁❂ t H f × H f × H f s R ( t ( ) ( ) ( ) ) T C R 0 ❀ ❃ ❂ ↔ S f s t ( ) ( ) R R s R ( t ) s R t Samples of ( ) 1 ❄ ❅ ❆ t 0 − − T T T T T T 2 2 3 4 b b b b b b EE456 – Digital Communications 7
Chapter 9: Signaling Over Bandlimited Channels Frequency-Domain Nyquist’s Criterion for Zero ISI It can be proved that (see textbook) the equivalent Nyquist’s criterion in the frequency domain is that the sum of s R ( f ) and all of its delayed copies, delayed by integer multiples of bit rate , is a constant! 1 1 2 S f + − − S R ( f S f S f ◆ ❑ ) ❍ ❊ ❍ ❊ R R R T T T ▲ ■ ❋ ❈ ❋ ❈ ▲ b ■ ❋ b ❈ ❋ b ❈ ▼ ❏ ● ❉ ● ❉ ❇ ❇ f 1 1 0 1 1 3 2 − − T T T T T T 2 2 2 b b b b b b ∞ k + S f ❖ P R T ❯ ◗ ❘ k =−∞ b T ❙ ❚ b f 1 0 1 − T T 2 2 b b 1 If W < 2 T b , or r b > 2 W ⇒ ISI terms cannot be made zero. 2 T b | , s R ( t ) = sin( πt/T b ) 1 1 2 T b , or r b = 2 W ⇒ S R ( f ) = T b over f ≤ | If W = ( πt/T b ) . 1 If W > 2 T b , or r b < 2 W ⇒ Infinite number of S R ( f ) to achieve zero ISI. EE456 – Digital Communications 8
Chapter 9: Signaling Over Bandlimited Channels 1 Pulse Shaping when W = 2 T b , or r b = 2 W S R ( f ) T b f 0 1 1 − T T 2 2 b b 1 A good approximation of a brick-wall filter 0.8 is not simple (e.g., requires a very long 0.6 filter if implemented digitally). s R ( t ) decays as 1 /t ⇒ if the sampler is 0.4 s R ( t ) not perfectly synchronized in time, 0.2 considerable ISI can be encountered. 0 −0.2 −0.4 −3 −2 −1 0 1 2 3 t / T b EE456 – Digital Communications 9
Chapter 9: Signaling Over Bandlimited Channels Raised Cosine (RC) Pulse Shaping (Industry-Standard) | f | ≤ 1 − β T b , 2 T b T b cos 2 � � �� πT b | f | − 1 − β 1 − β 2 T b ≤ | f | ≤ 1+ β S R ( f ) = S RC ( f ) = , . 2 β 2 T b 2 T b | f | ≥ 1+ β 0 , 2 T b s R ( t ) = s RC ( t ) = sin( πt/T b ) cos( πβt/T b ) = sinc ( t/T b ) cos( πβt/T b ) . 1 − 4 β 2 t 2 /T 2 1 − 4 β 2 t 2 /T 2 ( πt/T b ) b b 1 1 β =0 0.8 β =0.5 0.8 0.6 0.6 0.4 S R ( f ) s R ( t ) 0.2 0.4 β =0 β =1.0 β =1 0 0.2 −0.2 β =0.5 0 −0.4 −3 −2 −1 0 1 2 3 −1 −0.5 0 0.5 1 fT b t / T b EE456 – Digital Communications 10
Chapter 9: Signaling Over Bandlimited Channels With the rectangular spectrum 2 1 y ( t )/ V 0 −1 −2 0 1 2 3 4 5 6 7 8 9 10 11 t / T b With a raised cosine spectrum 2 1 y ( t )/ V 0 −1 −2 0 1 2 3 4 5 6 7 8 9 10 11 t / T b EE456 – Digital Communications 11
Chapter 9: Signaling Over Bandlimited Channels Eye Diagrams Eye diagrams are used to observe and measure (qualitatively) the effect of ISI. An eye diagram is obtained by overlapping and displaying multiple segments of the received signal (after the matched filter) over the duration of a few symbol periods. The wider the eye is opened, the better the quality of the samples under mismatched timing and AWGN. 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 y ( t )/ V y ( t )/ V 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −2 −2 −2.5 −2.5 0 1.0 2.0 0 1.0 2.0 t / T b t / T b Left: Ideal lowpass filter; Right: A raised-cosine filter with β = 0 . 35 . EE456 – Digital Communications 12
Chapter 9: Signaling Over Bandlimited Channels Eye Diagrams under AWGN with SNR = V 2 /σ 2 w = 20 dB 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 y ( t ) /V y ( t ) /V 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −2 −2 −2.5 −2.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t/T b t/T b Left: Ideal lowpass filter; Right: A raised-cosine filter with β = 0 . 35 . EE456 – Digital Communications 13
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